Effect of noise and perturbations on limit cycle systems

Physica D 50 (1991) 311-320 North-Holland

Effect of noise and perturbations on limit cycle systems Christian Kurrer and Klaus Schulten Department of Physics and Beckman Institute, Unit~ersity of Illinois at Urbana-Champaign, 405 N. Mathews, Urbana, IL 61801, USA

Received 15 March 1990 Accepted 24 September 1990 Communicated by A.V. Holden

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We tfien describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.

I. Introduction: Nonlinear dynamical systems in biology

e q u a t i o n s ( H H e q u a t i o n s ) to r e p r o d u c e the action potential of a squid n e u r o n . F i t z H u g h [7] proposed a t r a n s f o r m a t i o n to reduce the four-

M a n y life processes d e p e n d o n n o n l i n e a r con-

d i m e n s i o n a l H H e q u a t i o n s to a t w o - d i m e n s i o n a l

trois. E x a m p l e s are biochemical kinetics [1], b i o m o l e c u l a r evolution [2], m o r p h o g e n e s i s [3, 4], the circadian clock [5], a n d n e u r a l systems, e.g. a single nerve cell [6]. I n some cases, the u n d e r l y i n g n o n l i n e a r dynamical systems are well u n d e r s t o o d , for example in biochemical kinetics, in o t h e r cases a t t e m p t s are b e i n g m a d e to show that u n k n o w n control m e c h a n i s m s can be m o d e l l e d by r a t h e r simple n o n l i n e a r dynamics. In this p a p e r we w a n t to focus o n biological systems a n d o n c o r r e s p o n d ing models that exhibit excitable or oscillatory behaviour, mainly o n the dynamics of a single neuron. In g r o u n d b r e a k i n g work, H o d g k i n a n d Huxley [6] p r o p o s e d a set of four n o n l i n e a r differential

set of equations. T h e e q u a t i o n s arrived at are the so-called B o n h o e f f e r - v a n der Pol e q u a t i o n s (BvP e q u a t i o n s ) based on earlier work by van der Pol [8] a n d Bonhoeffer [9]. T h e d e t e r m i n i s t i c models ( H H a n d BvP) were successful in r e p r o d u c i n g the two states of a n e u r o n , firing at a c o n s t a n t rate and silence. However, a d d i t i o n a l stochastic processes [10-14] n e e d to be a d d e d to the BvP dynamics to a c c o u n t for a variable rate of firing that can vary c o n t i n u o u s l y b e t w e e n silence, i.e. firing at a rate of 0 kHz, a n d firing at a rate of a b o u t 1 kHz. In the following a description of stochastic d y n a m i c systems suggested by van K a m p e n [15] will be applied to analyze the BvP model in the

0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

312

('h. Kurrer and 1<2.Schulten / Limit cyc/e vvstems

proximity of a neuron's limit cycle. We will demonstrate that local properties of the dynamics near the limit cycle have a great impact on the way that the corresponding systems can be influenced by random noise or by interactions with other oscillating systems, In section 2 the Fokker-Planck equation for a neuron subject to stochastic forces and the van Kampen solution in the limit of small fluctuations are introduced. In section 3 the van Kampen solution is analyzed. In section 4 implications of our results for large systems of interacting nonlinear oscillators, e.g. neurons in neural networks, are discussed.

2. Mathematical framework 2.1. Deterministic systems Our investigations will be based on a system described by the nonlinear differential equations (BvP equations)

2.2. Stochastic systems In order to show that the limit cycle of the BvP model exhibits different stability characteristics along different sections one can apply various perturbations during a revolution of the system around the limit cycle and monitor the response. A more systematic approach for such investigation is to apply noise to the system and to investigate the dynamics of a large ensemble of BvP oscillators. For the sake of simplicity we apply isotropic Gaussian white noise r/(t), characterized by

(,7(t)) =0, (T~(tl)T~(t2))=

2j = FI( x l , x : )

~

la(tl --t2).

(2)

Such noise added to the BvP equations yields the stochastic differential equations

A'2 = F 2 ( x 1 , x 2 )

= (a - x 1 - b x 2 ) / c

an unstable stationary point and exhibits limit cycle behaviour; in the range -0.3465 < z < 2.0 the system has a stable stationary point. In the following, we will use a value of z = 0.4, for which the stationary point is unstable such that all trajectories eventually lead into a limit cycle.

(1)

2 1 - FL( x l , x 2 ) + ~7~(t ) , introduced by FitzHugh [7]. In this model x 1 represents essentially the voltage of a neuron and x 2 can be identified with the conductivity of the potassium channels of the neural membrane, z is an essential control parameter of the system and in the range - 0.6 < z < 2.0 represents physiologically possible excitation of the system due to transmembrane currents [7, 14]. The BvP equations provide a prototype for two-dimensional nonlinear dynamical systems that show excitable and oscillatory behaviour. We assume the parameters a = 0.7, b = 0.8, c = 3.0, which are widely used in the literature [7, 14]. Phase plane analysis [7, 14] of the stationary point of eq. (1) shows that in the range - 0.6 < z < - 0.3465 the system has

2~=Fz(xl,x2)

(3)

+ r/z(t).

An ensemble of systems governed by eqs. (3) is described by the density p ( x , t) which gives the probability to find a system at position x = ( x j , x 2 ) T at time t. The evolution of p ( x , t ) is governed by the Fokker-Planck equation [16]

"¢p(x, t)

=(_

~.aiFi(x)+ 1213l~_.et~)p(x,t). i

(4)

i

We want to consider the solution of eq. (4) for

Ch. Kurrer and K. Schulten / Limit cycle systems

the initial condition #1

(5)

p ( x , t = O) = 6 ( X - X o ) .

Van Kampen [15] analyzed solutions of (4), (5) in the limit of weak noise. In this limit, p(x,t) should be peaked around the deterministic solution Xdet of eqs. (1). Van Kampen [15] proposed to expand the Fokker-Planck equation in terms of deviations from the deterministic solution, scaled by the square root of the noise amplitude /3 defined in eq. (2)

y( t ) =/31/2[x

- Xdet(t)]

(6)

.

The scaling f a c t o r / 3 1 / 2 in eq. (6) allows to eliminate the explicit occurrence of /3 in the Fokker-Planck equation. Replacing the force field F(x) by OF

F( x) = F( Xdet) + ( - ~ ) xdct " /3

l/2y

(7)

313

appear in eq. (8) since it has been eliminated by the substitution described in eq. (6). In fact, noise will be represented by a constant term (eq. (10)) independent of the actual amplitude of the noise. The solution of the Fokker-Planck equation (8) according to ref. [15] is given by a Gaussian probability distribution,

p(y,t)

1 e x p ( - ~yT • ~ - ' "y), 2'rrDx/D~----

(11)

with =~j(t) = (yi(t) yj(t)). The matrix ---(t) describes the second moments of the Gaussian probability distribution, i.e., size and orientation of the distribution in phase space. The overall width of the distribution p(y, t) is

~ = ~ .

(12)

and, thus, omitting terms of higher order i n / 3 - 1/2, eq. (4) can be written in terms of the new coordinates y(t) as

---(t), together with the position of the deterministic solution Idet(t) governed by eq. (1), provides a complete description of the probability distribution under the approximation made in eq. (7). ---(t) is determined by the equation of motion [15]

Otp(y,t ) = _ Y ' ~ A i j ( t ) ~ i y j p ( y , t )

--

i,j

1

~)2

+~ ~.~Bij~p(y,t),

(8)

l,J

where

Aij(t) = ~ i

ogi Xdet(t)

Bij = aij"

'

(9) (10)

The amplitude /3 of the noise does no longer # l T h r o u g h o u t this paper, we use the value of x 0 = (x 1, x 2) = (1.11, - 0 . 3 2 ) , which lies on the limit cycle in the proximity of the stationary point, as starting point for any numerical calculations.

=

A(t)--

+

~.AT(t)

+

B

(13)

and by the initial conditions ="ij(t = 0 ) = 0 for i,j = 1,2, which follows from eq. (5). The elements of A(t), defined in eq. (9), take on positive as well as negative values and describe the expansion, contraction and rotation of the probability distribution. The elements of B, defined in eq. (10), are constant and positive and describe a diffusive broadening of the distribution. The description outlined is valid as long as the linearization introduced in eq. (7) holds, i.e. as long as the distribution extends over a small range. In the long time limit, this approximation does no longer apply. The distribution eventually will reach a stationary state in which it is smeared out over the whole limit cycle [17].

314

Ch. Kurrer and K. Schulten /Limit cycle systems

3. Van K a m p e n analysis of the effect of noise on the BvP limit cycle

2.5 all 0.0

In this section we will investigate the effect of noise on the limit cycle of a BvP system. For this purpose we will assume first that the motion along the limit cycle and normal to the limit cycle are uncoupled, and that noise affects these motions separately. The effect of noise is easier to understand since the corresponding one-dimensional stochastic dynamical systems can be analyzed without resorting to numerical solutions. We wilt then present the solution of eq. (13) for coupled tangential and normal motion along the limit cycle.

-2.5 -5.0 -7.5 -10.£

0.2

0.4 0.6 time

0.8

1.0

Fig. 1. Derivative at~ of the longitudinal force Fl in the direction tangential to the limit cycle during one revolution along the limit cycle. The time axis is labeled in units of the limit cycle period T. to the limit cycle, eq. (14) reduces to

3.1. Effect o f noise in direction along the limit cycle

(ll = 2aHsCtx + b,

For the analysis presented in sections 3.1, 3.2 we transform from yl(t), y2(t) coordinates to coordinates yl(t) tangential and y , ( t ) normal to the limit cycle at the position Xdet(t) of the deterministic motion. In such a coordinate representation eq. (13) reads

which describes the expansion and contraction of the longitudinal distribution during its motion along the limit cycle. The solution of (15) yields the width ~rl of the distribution when motion is confined to the limit cycle

dtts~nl

+

~c l

,,n)=laL

~,n

~c

\a,t

at.

annJ~,,

ann

+

~,n

0

~rl = ~ .

t

b "

(14)

H e r e we have defined -=tt = ~,, ~ln = ~ln, etc. and A , = a H, A t , = at,, etc. These equations neglect terms due to the time dependence of the transformation. Since the equations will serve only for a qualitative analysis of the behaviour of the diagonal elements of ~ ( t ) the neglect of 'Coriolis-type' terms is not essential. The value of b is 1, if the system is subjected to noise, and 0 in the absence of noise. First, we want to consider separately the dynamics in the two directions normal and tangential to the limit cycle. In the direction tangential

(15)

(16)

The solution is determined through the values of all = 8Ft/Oy l evaluated at points Xdet(t) along the limit cycle. The index ! refers to the direction tangential to the limit cycle, t ~ [0, T[ denotes the time which serves here as a p a r a m e t e r to define the locations on the limit cycle, T being the period of the limit cycle. We will also allow t values larger than T which correspond then to more than one revolution around the limit cycle. Fig. 1 presents the values of a , for one revolution around the limit cycle. The variation of a , demonstrates clearly that the dynamics along the limit cycle is not uniform. One can discern regions with positive a , (expansion of width ~rt) and with negative a , (contraction of width ~rt). The question arises which of the contributions of a~t, i.e. the positive or the negative ones, are more

Ch. Kurrerand K. Schulten / Limit cycle systems prevailing. To answer this question we consider motion a r o u n d the limit cycle first in case of a system without noise, i.e. for the case b = 0 in (15). Integration of (15) yields

315

lOO 80 60 40

~lt(t + T ) [ rt+T =~ll(t)exp~2Jt aH(t ' )

dt') .

20

(17)

o

0.2 T o evaluate the integral in the e x p o n e n t we will introduce the path e l e m e n t ds along the limit cycle which is related to dt by dt = F~-1 ds. With the identities

aa( t ) dt = ( d F l / d s ) ( 1 / F t ) ds = d(log Fi) one obtains

expl2[t2a,,(t,r)dt,j(

~ = [ Fl(t,)1 z

(18)

F o r t 2 = t~ + T this expression is unity and, hence, ~tl(t + T ) = sell(t). In the noise-free case this cyclic p r o p e r t y of ~tl(t), of course, is to be expected. T h e derivation shows that the contribution of at~ in eq. (15) along a c o m p l e t e revolution around the limit cycle balances contracting and expanding contributions. This implies, however, that for a system subjected to noise over one or over several revolutions a r o u n d the limit cycle the noise term will d o m i n a t e ~ll(t) and will induce a diffusion-type broadening. T o study such b r o a d e n i n g we consider now the integration of (15) for nonvanishing b. Integration of (15) yields

~:tt(to + t)

0.4 0.6 time

0.8

1.0

Fig. 2. Longitudinal and normal widths ~l (--) and % (. • • ) of the probability distribution during one limit cycle period. The time axis is labeled in units of the limit cycle period T. o'. has been multiplied by a factor 10 to be visible on the same scale. T h e resulting width o-t over one revolution a r o u n d the limit cycle is p r e s e n t e d in fig. 2. O n e can recognize two segments along the limit cycle during which the width increases and decreases. T h e beginning of these two s e g m e n t s coincide with those parts of the limit cycle for which all assumes positive values. O n e also discerns in fig. 2 that the p r e s e n c e of noise leads to a small, but significant residual width after the revolution a r o u n d the limit cycle is completed. To d e m o n strate how this residual width develops over a longer time period we p r e s e n t in fig. 3 the width tr t over 20 revolutions a r o u n d the limit cycle. T h e results show an increase of tr t proportional to ~-. This diffusion-like behaviour can be derived from the solution (19) taken for t = NT. By m e a n s of

125 100

GI

75

25

= [ " t ( t ° ) + b f t °•+o t d t ' e x p (

- 2 ftot'dt'all(t") )1

0 4

8

I

I

12

16

20

time

×exp(2/ k

to+t

"t 0

t

t

dt att(t )). /

(19)

Fig. 3. Longitudinal width o-~= ~:~/2 of the probability distribution during 20 limit cycle periods.

316

Ch. Kurrer and K. Schulten /Limit cycle systems

(18) we obtain

2.5

~ll( to + N T ) N-Ifto+(j+l)r (Ft(to+jT) = ~H(t) + b j=0 Y'~ "to+iT El(t ) dt

2.

ann o.o ~-2.5 -5.0

(20) Since Ft(t) is cyclic, i.e. Fl(t + T)=Fl(t), the contributions in the sum are all identical and the final expression for ~l(to + N T ) is

~,z(to + N T ) = ~l,(t) + NTbF,2(to)

( F,-2 (t)) 7" (21)

In this expression ( F I - 2 ( t ) ) T d e n o t e s a time average which is equal to the space average (L/T)(F~-3(s))L where L is the length of the limit cycle. F r o m eq. (21) follows

0.,( t o + NT) = 1//o'/2(t0)

-F NTbF,2( to) ( F , - 2 ( t ) ) r

-7.5 -10.0

0.2

0.4 0)6 time

0.8~.0

Fig. 4. Derivative ann of the normal force F, in the direction orthogonal to the limit cycle during one revolution along the limit cycle. The time axis is labeled in units of the limit cycle period T. the limit cycle is stable against perturbations in the normal direction. This stability, in fact, is so strong, that contraction due to ann and diffusive b r o a d e n i n g due to b induces the distribution to follow (approximately) adiabatically the equilibrium width

(22) which clearly exhibits the ~- d e p e n d e n c e (t = N T ) seen in fig. 3.

3.2. Effect of noise in direction normal to the limit cycle In the direction normal to the limit cycle, eq. (14) reduces to ~'nn = 2ann~:nn + b.

%(t) ~

i

b

(24)

_2am,(xd~t(t))

along those segments of the limit cycle where a .... is negative. This behaviour is clearly reflected in the long-time behaviour of 0.,(t) presented in fig. 5; the width 0.n(t), after a brief transient bchaviour during the first limit cycle period, shows a cyclic behaviour at later times, i.e. 6rn(t) averaged over consecutive limit cycles remains constant.

(23)

Solution of this equation yields the width of the distribution normal to the limit cycle defined as 0.~ = ~ . Fig. 4 presents the values of a,n during one revolution a r o u n d the limit cycle, a,1,1 is f o u n d to be negative over most, but not all parts of the limit cycle. Negative values of ann correspond to contraction of the normal distribution, i.e. trajectories near segments of the limit cycle with a,~ < 0 are attracted to the limit cycle. T h e d o m i n a n c e of negative ann values implies that the average T - l f / + T d t ' a ~ , ( t ') is negative and that

3 13n 2

o 4

8

2

16

20

tim, Fig. 5. Normal width o-n ~tl,~2 of the probability distribution during 20 limit cycle periods. =

Ch. Kurrerand K. Schulten ,/Limit cycle systems The increase and decrease of ~,(t) according to eq. (24) reflects the change of the local ann values along the limit cycle. During the two short segments where ann is positive, the width briefly expands exponentially, to be contracted again along segments with ann < 0. The variable behaviour of o-l and o-n along a limit cycle revolution demonstrates strikingly that the stability of the limit cycle against noise and other perturbations varies along its path. The occurrence of exponential growth and peaks in the width o-n of the distribution at certain points of the limit cycle implies that the Brownian trajectories at these points will show great variability in their deviation from the limit cycle to either side. Small disturbances at the beginning of an exponential growth period (an, > 0) have a large effect on the trajectories. The strong variation of o-l along the limit cycle and its diffusive behaviour on a time scale of many limit cycle periods implies that the phase of the system along the limit cycle can be affected through noise, and that the effect of noise depends much on the position on the limit cycle where it acts on the system. We want to demonstrate in the next subsection that the effects of noise in the direction tangential and normal to the limit cycle are actually strongly coupled such that the behaviour of both ~rI and ~r. is most relevant for judging how noise or external perturbations will influence an oscillating BvP system.

3.3. Effect o f noise on coupled tangential and normal motion along the limit cycle Up to now we treated the two directions of diffusion normal and tangential to the limit cycle separately. We now want to focus on effects that arise from the coupling of normal and tangential diffusion in eq. (13). This coupling is effected through the nondiagonal elements of the ,6, matrix in eq. (13), which had neither been considered in eq. (15) nor in eq. (23). We will deal with eq. (23) in the y~, Y2 representation, i.e. the error

317

500 I ~ ~(112)

(tr=)

400 300 200 100 0 [ 4

8

12

16

20

time

Fig. 6. Overall width (Tr _--)i/2 of the probability distribution during 20 limit cycle periods without coupling between longitudinal and normal diffusion, i.e. eqs. (15) and (23) were used to evaluate the diagonal elements of -- separately.

from the time dependence of the Yl, Y2 ~ Yl, Yn transformation does not arise. It is intuitively clear that if the phase point is disturbed in a direction normal to the limit cycle, a trajectory results that eventually reaches the limit cycle again, but its phase point, in general, is not found at exactly the same position as another phase point that started at the same position on the limit cycle but was not disturbed. Because of this difference of positions between the two phase points, fluctuating forces normal to the limit cycle contribute to diffusive broadening in the longitudinal direction of the limit cycle and, as will be shown shortly, can enhance significantly the effect of noise on the phase along the limit cycle. Figs. 6 and 7 compare the evolution of the overall width o- of the distribution defined in eq. (12) during propagation over 20 limit cycle periods with and without coupling of longitudinal and normal motion. The results presented in fig. 6 have been obtained by integrating separately eq. (15) and eq. (23) and then evaluating ~(t) = ~ + bC:nn . The results presented in fig. 7 have been obtained by integration of the matrix differential equation (13) and evaluation of ~r(t) according to the definition (12). Comparison of figs. 6 and 7 shows that the coupling of longitudinal and normal motion increases the overall width by a factor of approximately four over the width o-(t) = f ~ u + ¢.n of the uncoupled motion.

318

Ch. Kurrer and K. Schuhen / Limit o'cle systems

500 ! ttr-~ W2~ I -'J 4001

3ool 2OO ! 100 I

0 4

8

I

I

12

16

20

time Fig. 7. Overall width (Tr .~)1/2 of the probability distribution

during 20 limit cycle periods with coupling between longitudinal and normal diffusion, i.e. eq. (13) was integrated to obtain the matrix --. Further analysis of the results showed that this increase is mainly due to an increase of the tangential component of ~ ( t ) . The results in figs. 6 and 7 demonstrate that noise acting on the motion normal to the limit cycle contributes strongly to phase diffusion.

4. Concluding remarks and outlook We have demonstrated above that the influence of noise and, hence, of any other external perturbation on a nonlinear oscillator described by the BvP equations varies strongly along the limit cycle. For this purpose we have employed the van K a m p e n approximation to describe the width of a distribution circling the limit cycle. The van K a m p e n approximation characterizes the stability of nonlinear oscillators against phase diffusion and deviations from the limit cycle. Although the approximation becomes exact only in the limit of weak noise, it should often also yield information about responses which are characterized by large deviations from the unperturbed oscillator motion. This situation is similar to the case of the celebrated linear stability analysis for nonlinear dynamical systems which applies strictly only in a small neighborhood of a fixpoint but nevertheless often allows conclusions about global dynamics patterns.

The properties of single nonlinear oscillators subject to external perturbations are significant in view of the interesting behaviour of systems of coupled nonlinear oscillators. Such systems can be characterized often as attractor networks, in that sets of initial conditions can be considered basins of attraction which lead the network to a small number of final states, the most prominent among them the state in which the oscillators are all synchronized. The results of our p a p e r should be pertinent to neural networks viewed as coupled nonlinear oscillators. We like to argue that in such systems one has to take account of the structure of the limit cycle of neurons. The latter operate often in a regime of control parameters near the transition between oscillatory and stationary behaviour: in the oscillator regime neurons are sufficiently and continuously excited that they exhibit a constant rate of firing, in the stable stationary state regime neurons behave as threshold units which fire only once after a sufficient deviation from the stationary point. We like to explain now to what extent a limit cycle of a BvP neuron becomes more structurally unstable as the control p a r a m e t e r z approaches the critical value z c = -0.3465 at which the transition to a stable stationary point behaviour takes place. Structural instability of the system for z = zc, z < z~ implies that a neuron reacts sensitively to small perturbations by shifting its firing phase considerably. This sensitivity, observed both for fluctuations tangential and fluctuations normal to the limit cycle, can be readily explained. As z approaches z c, the stationary point moves closer and closer to the limit cycle. The motion of the phase point on the limit cycle near the stationary point will slow down, whereas the motion in regions of the limit cycle further away from the stationary point is affected only insignificantly. The phase point, therefore, spends more and more of the limit cycle period in a relatively small segment of the limit cycle, i.e. in proximity to the stationary point. By perturbing the phase point in that region of the limit cycle, one can shift its phase by a large amount.

Ch. Kurrer and K. Schulten / Limit cycle systems

X2 O.

-2

-3

'0

Xl

3

Fig. 8. L i m i t cycle ( - - ) and s t a t i o n a r y point (e) in the p h a s e p l a n e of the BvP m o d e l for z = - 0.4. On the limit cycle, the regions most sensitive to t a n g e n t i a l (. • - ) and n o r m a l ( - - - ) p e r t u r b a t i o n have b e e n encircled.

In the direction normal to the limit cycle, perturbations also have an increasing impact on the phase of the limit cycle. The reason is that ann = OFn/Oxn,which, as shown earlier, is a measure of the local stability of the limit cycle, generally increases in the proximity of the stationary point as the stationary point approaches the limit cycle. Even small perturbations normal to the limit cycle can then lead to large deviations of the trajectories of the phase point away from the limit cycle and eventually translate into a large phase shift. (For an extension of the definition of phase to regions in the proximity of the limit cycle, see e.g. ref. [18].) Fig. 8 illustrates the sensitive areas on the limit cycle. It shows areas, characterized by a local minimum of ch, that are sensitive to longitudinal perturbations, and areas, characterized by positive ann, that are sensitive to normal perturbations. We like to propose that structural instability plays an important role in the synchronization of nonlinear oscillators. Many experimental studies suggested that fast synchronization of neurons in the visual [19, 20] and olfactory [21] cortex is a key element in the process of feature detection. Most former theoretical investigations into the self-synchronization of nonlinear oscillators [18, 22-24] assumed structurally stable limit cycles as components of a large ensemble; such oscillators,

319

however, allow only gradual synchronization. Furthermore these investigations focused on the question to what extent the degree of synchronization of a population of oscillators can be controlled through the strength of the inter-oscillator coupling. The investigations would imply that neural systems achieve synchronization through variation of synaptic interactions. However, experimental evidence for short time variation of the synaptic strength of neurons is scarce and disputed. We like to suggest that the transition to a synchronized state of neurons is not mediated by varying the strength of synaptic connections, but rather by changing the susceptibility to synchronization of the cells by effectively altering the z value, which in the BvP model represents the "excitation" of the neuron, nearer to its critical value. Such change can be achieved through systematic input of neurons, e.g. through input from the retina to a subset of neurons in the optical cortex. If neurons representing a particular feature, e.g. optical flow into a certain direction and of a particular magnitude, are excited in an area of the image corresponding to an object, they may receive sufficient excitation to be sensitized such that lateral interactions specific for the subpopulation of neurons coding the respective feature suffice to synchronize firing. Such linking of neurons may, for example, explain filling-in capabilities or figure-ground separation. The importance of the above outlined structural instability for the fast synchronization of neurons is supported by the observation that neurons in the olfactory and visual cortex alternate between excitable and self-oscillatory behaviour and, thus, seem to operate in the dynamical regime where the z value in the BvP model is close to its critical value.

Acknowledgements This work has been supported by the University of Illinois at Urbana-Champaign and by a grant of the National Science Foundation DMR

320

Ch. Kurrer and K. Schulten / Limit cycle systems

89-20538 (through the Materials Research Laboratory of the University of Illinois),

Note added in proof Since the submission of this article various groups have reported investigations on the capacity for figure ground separation through synchronized firing, see for example ref. [25].

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